Representative Scott Rigell of Virginia says he plans to vote for the Libertarian Party’s presidential ticket, becoming the first member of Congress to express support for Gary Johnson’s third-party campaign.

In an interview on Friday, Mr. Rigell, a Republican, said he had settled on Mr. Johnson, the former governor of New Mexico, as the best option available.

“I’ve always said I will not vote for Donald Trump and I will not vote for Hillary Clinton,” Mr. Rigell said. “I’m going to vote for the Libertarian candidate.”

Mr. Rigell is the second Republican member of Congress this week to back a candidate other than Mr. Trump in the presidential race; the other, Representative Richard Hanna of New York, endorsed Mrs. Clinton. Both Mr. Rigell and Mr. Hanna are retiring from Congress at the end of their current terms.

But Mr. Rigell’s support for the Libertarian ticket may be more portentous: Mr. Johnson and his running mate, former Gov. William F. Weld of Massachusetts, have been seeking to woo disaffected Republicans who view Mr. Trump as unacceptable. Should additional Republican officeholders follow Mr. Rigell in backing the third-party ticket, it could help Mr. Johnson gain credibility as an alternative choice.

A former car dealer who represents a moderate district on the Virginia coast, Mr. Rigell said he expected more congressional Republicans and local Republican officials to break openly with the party’s presidential candidate as the election draws nearer.

Many Republican candidates have asked him for advice, Mr. Rigell said, about separating themselves from Mr. Trump’s campaign. Mr. Rigell has been saying since the winter that he could not support Mr. Trump.

“When their own conscience is seared by some statement that Trump has made, I have encouraged them to be direct and also, in a timely manner, repudiate what he said,” Mr. Rigell said. “People will respect it if you have a reason and you put it out there.”

Mr. Rigell said he still considered himself a Republican. But that would change, he said, if the Republican Party were to become synonymous with Mr. Trump and his ideas in a lasting way.